# Solution: 2014-07 Subsequence

Let $$a_1,a_2,\ldots$$ be an infinite sequence of positive real numbers such that $$\sum_{n=1}^\infty a_n$$ converges. Prove that for every positive constant $$c$$, there exists an infinite sequence $$i_1<i_2<i_3<\cdots$$ of positive integers such that $$| i_n-cn^3| =O(n^2)$$ and  $$\sum_{n=1}^\infty \left( a_{i_n} (a_1^{1/3}+a_2^{1/3}+\cdots+a_{i_n}^{1/3})\right)$$ converges.

The best solution was submitted by 장기정(2014학번). Congratulations!

Alternative solutions were submitted by 정성진 (+3), 이종원 (+2), 이영민 (+2), 황성호 (+2), 김경석 (+2), 채석주 (+1). Incorrect solutions were submitted by B.H.J., P.K.H.

GD Star Rating